How do I create a smooth curve along an edge on this model?
I would use an actual curve object in combination with a Screw Modifier.
First create a curve that will be one of four curve edges of your final object. Then in Edit Mode make it the shape you want, but also move it away form the origin in a way that will make the origin the center of your final object. (note the orange dot/origin in the gif image)
Then put a Screw Modifier on the curve object. Set the Axis to X, the Angle to 360, and the Steps to 4. Now if you manipulate the curve, all four curved edges change accordingly.

When you are happy with the shape press Alt + C and choose Mesh From Curve/Meta/Surf/Text, which changes your object to a mesh and applies the modifier.
Since the edge curve in your example image highlighted in white actually splits into two edges at some point, you might want to model this part manually as proposed by moonboots. You can select part of the edges of the curve, up to where the split would be split, and press Ctrl + B to bevel the curve and create the split. (in this example the top part would also have to be scaled down manually to make the curve flow nicely)